Abstract
The recently discovered kagome superconductor CsV_{3}Sb_{5} (T_{c} ≃ 2.5 K) has been found to host charge order as well as a nontrivial band topology, encompassing multiple Dirac points and probable surface states. Such a complex and phenomenologically rich system is, therefore, an ideal playground for observing unusual electronic phases. Here, we report anisotropic superconducting properties of CsV_{3}Sb_{5} by means of transversefield muon spin rotation (μSR) experiments. The fits of temperature dependences of inplane and outofplane components of the magnetic penetration depth suggest that the superconducting order parameter may have a twogap (s + s)wave symmetry. The multiband nature of superconductivity could be further supported by the different temperature dependences of the anisotropic magnetic penetration depth γ_{λ}(T) and upper critical field \({\gamma }_{{{{{\rm{B}}}}}_{{{{\rm{c}}}}2}}(T)\). The relaxation rates obtained from zero field μSR experiments do not show noticeable change across the superconducting transition, indicating that superconductivity does not break time reversal symmetry.
INTRODUCTION
The kagome lattice materials, consisting of a twodimensional lattice of cornersharing triangles, have drawn considerable attention in recent years^{1,2,3,4}. Their electronic structure is characterized by a dispersionless flat band, whose origin lies in the innate kinetic frustration of the kagome geometry, and a pair of Dirac points^{5}. Such flat bands, with a high density of electronic states, are generally perceived to quench the kinetic energy and to induce correlated electronic phases when found close to the Fermi level^{6,7}, as illustrated by the recently discovered superconducting twisted bilayer graphene^{5}. The inherent geometrical frustration of kagome systems can be employed to carefully tune their properties, thus aiding in the search of superconductors (SC) with nonphonon mediated pairing mechanisms^{8}. A recent example of a kagome superconductor with unconventional coupling is LaRu_{3}Si_{2}^{8}. Here, the correlation effects from the kagome flat band, the van Hove points on the kagome lattice, and the high density of states from the narrow electronic bands were proposed as key factors for achieving a relatively high transition temperature T_{c} ≃ 7 K.
Following the recent discovery of the AV_{3}Sb_{5} (A = K, Rb, Cs) family of kagome materials^{9}, a slew of interesting and exotic effects have been observed: giant anomalous Hall conductivity^{10,11,12}, magnetoquantum oscillations^{10,13}, topological charge order^{14,15,16,17,18,19}, orbital order^{20}, and superconductivity^{21,22,23,24}. Featuring a kagome network of vanadium atoms interwoven with a simple hexagonal antimony net, the normal state of CsV_{3}Sb_{5} was described as a nonmagnetic Z_{2} topological metal^{21,22}. Furthermore, the observation of CDW order in the normal state of all members of AV_{3}Sb_{5} kagome family has generated significant theoretical and experimental interest. Namely, topological chiral charge order has been reported in AV_{3}Sb_{5} (A = K, Rb, or Cs)^{14,16}. In KV_{3}Sb_{5}, direct evidence for timereversal symmetry breaking by the charge order was demonstrated using muon spin rotation^{25}.
Regarding superconductivity, a strong diversity in the SC gap symmetry is reported in AV_{3}Sb_{5} family. Proximityinduced spintriplet pairing was suggested for K_{1−x}V_{3}Sb_{5}^{26}. For CsV_{3}Sb_{5}, in particular, there are significant differences in concluding the superconducting gap structure. For instance, multiband superconductivity with signpreserving order parameter was reported by means of scanning tunneling microscopy (STM) measurements at ultralow temperature^{27}, and magnetic penetration depth measurements using tunnel diode oscillator techniques^{28}. Contrarily, a nodal type superconducting gap symmetry is proposed through thermal conductivity measurements^{29}. Finally, reentrant superconductivity and double SC domes were found under pressure^{30,31,32}. From a theoretical perspective, several scenarios for electronically mediated, unconventional superconductivity have been discussed^{33}. The AV_{3}Sb_{5} electronic bands exhibit van Hove singularities close to the Fermi energy—an electronic structural motif shared with other systems, such as the cuprate superconductors or Sr_{2}RuO_{4}. A particular feature of the kagome lattice, however, is a sublattice interference mechanism^{34}, by which the Bloch states near each van Hove point are supported on a distinct sublattice. This promotes the relevance of longrange interactions and unconventional pairing states.
To explore unconventional aspects of superconductivity in CsV_{3}Sb_{5}, it is critical to measure the superconducting order parameter on the microscopic level through measurements of the bulk properties. Thus, we focus on muon spin rotation/relaxation (μSR) measurements of the magnetic penetration depth^{35} λ in CsV_{3}Sb_{5}. λ is one of the fundamental parameters of a superconductor, since it is related to the superfluid density n_{s} via:
where m^{*} is the effective mass. Most importantly, the temperature dependence of λ is particularly sensitive to the topology of the SC gap: while in a fully gapped superconductor, \({{\Delta }}{\lambda }^{2}\left(T\right)\equiv {\lambda }^{2}\left(0\right){\lambda }^{2}\left(T\right)\) vanishes exponentially at low T, whereas in the case of a clean system with line nodes it shows a linear Tdependence.
RESULTS
Sample characterization
To determine sample purity, Xray diffraction experiments were performed on fluxgrown crystals. The powder diffraction pattern of ground crystals can be well fitted using the structure of CsV_{3}Sb_{5} and the fitted lattice parameters are a = 5.50552(2) Å and c = 9.32865(3) Å, close to the previous results^{22}. Additionally, to test the single crystallinity of the samples and determine the orientation for the μSR experiments, Xray Laue diffraction was performed on the single crystal shown in Fig. 1a, whose diffraction pattern is displayed in Fig. 1c. The crystal was easily aligned; the hexagonal symmetry of the abplane is clearly visible from the single crystal, and the crystals grow with the caxis aligned along the thin direction of the crystal. The diffraction pattern collected in Fig. 1c was analyzed with the OrientExpress program^{36} and the orientation was confirmed to be along the crystallographic caxis.
The superconductivity of the samples was confirmed by magnetization [Fig. 1d] and resistivity [Fig. 1f] experiments, which show a diamagnetic shift in the sample concurrent with the onset of zeroresistivity at T_{c} ≃ 2.7 K which is slightly higher than the compared to the T_{c} (= 2.5 K) value obtained from magnetization measurements. This is most likely due to very tiny filamentary superconducting channels. The onset of charge order is visible in the magnetization measurements in Fig. 1e, corresponding to the anomaly at T_{co} ≈ 95 K. There is also a slight change in slope visible in the resistivity data presented in Fig. 1f, which occurs at the onset of the chargeordered state.
Anisotropy in magnetic penetration depth and superconducting gap structure
Two sets of TFμSR experiments were carried out in the fieldcooled state, with the external magnetic field applied parallel to the c − axis, and parallel to the ab (kagome) plane. In both cases the muon spin was perpendicular to the applied field. Note that, for an applied field parallel to the caxis, the screening currents around the fluxline cores flow in the abplane. This allows us to determine the socalled inplane component of the magnetic penetration depth λ_{ab}. The TFμSR timespectra collected with an external field B_{ext} = 10 mT applied parallel to the caxis above (T = 5 K) and well below (T ≃ 0.27 K) the superconducting transition temperature T_{c} ≃ 2.5 K are shown in Fig. 2a. The corresponding Fourier transforms of the TFμSR data, representing the magnetic field distribution P(B), are shown in Fig. 2b. The insets in Fig. 2a and Fig. 2b, respectively, represent the geometry of the experiment and the schematic distribution of the magnetic fields within the isotropic fluxline lattice (FLL) with the two components of the magnetic penetration depth, namely λ_{a} and λ_{b} being equal: λ_{a} = λ_{b} = λ_{ab}. The TFμSR timespectra and the corresponding Fourier transforms collected with B_{ext} = 10 mT applied along the kagome plane are presented in Fig. 2c and d, respectively. With the field applied along the abplane, the screening currents around the vortex cores flow along the abplane and caxis, thus implying that in a set of experiments with B∥ab, λ_{ab,c} can be determined. Note that, due to the anisotropy, λ_{c} is longer than λ_{ab}, which leads to an elongation of the vortex lattice along the c direction [see inset in Fig. 2d].
The formation of the fluxline lattice (FLL) in the superconductor leads to a nonuniform magnetic field distribution between the vortices [see insets in Fig. 2b and d]. The strong damping of the TFμSR timespectra [Fig. 2a and c] and the corresponding broadening of the Fourier transform [Fig. 2b and d] represent exactly this effect. Note that the measured distribution of the magnetic fields in the superconducting state becomes asymmetric, as expected for a wellarranged FLL. All the characteristic features, as e.g., the cutoff at low fields (\({B}_{\min }\)), the peak due the saddle point between two adjacent vortices (B_{sad}), and the long tail towards high fields, related to the regions around the vortex core (\({B}_{\max }\)), are clearly visible for B_{ext}∣∣c. However. the asymmetric shape is not observed in case of B_{ext}∣∣ab for two reasons: (i) Long λ_{c}: long λ leads to smaller field variation within the vortex lattice. (ii) High value of the nuclear moment contribution σ_{nm}: since the asymmetric line shape caused by the formation of the vortex lattice needs to be convoluted with σ_{nm}^{37}, which effectively leads to a smearing out the characteristic features of the vortex lattice. The locations of \({B}_{\min }\), B_{sad}, and \({B}_{\max }\) are shown in the contour plot in the inset of Fig. 2b. To account for the field distribution P(B), the timedomain spectra were analyzed using a skewed Gaussian (SKG) function, which represents the simplest distribution accounting for the asymmetric lineshape (see Suppl. Mater. for a detailed description of the function).
The parameters obtained from the fits are presented in Fig. 2e–g. Figure 2e shows the temperature dependence of the square root of the second moment M_{2} (see Suppl. Mater. for detailed calculations) which corresponds to the total depolarization rate σ for two field orientations. Below T_{c}, the relaxation rate σ starts to increase from its normalstate value due to the formation of the FLL and saturates at low temperatures. The normalstate muon depolarization rate is mostly due to the nuclear magnetic moments and, for CsV_{3}Sb_{5}, it has different values for the two field orientations, \({\sigma }_{{{{\rm{nm}}}}}^{  c}\) = 0.165(3) μs^{−1} and \({\sigma }_{{{{\rm{nm}}}}}^{  ab}\) = 0.287(3) μs^{−1} [shown by horizontal lines in Fig. 2e]. As shown in Fig. 2f, for both field orientations the first moment—which represents the internal field (B_{int})—shows a clear diamagnetic shift below T_{c}, as expected for a typeII superconductor. Note that the T_{c} value estimated from μSR experiments agrees with that determined from magnetization measurement, so that \({T}_{{{{\rm{c}}}}}^{\mu {{{\rm{SR}}}}}={T}_{{{{\rm{c}}}}}^{{{{\rm{\chi }}}}}\simeq 2.5\) K.
The asymmetric line shape of the field distribution P(B) is characterized by the third moment (M_{3}) of the field distribution (see Suppl. Mater. for the calculations) and has three characteristic fields: \({B}_{\min }\), \({B}_{\max }\), and B_{sad}. More accurately, the asymmetry of the line shape is described by its skewness parameter \({\alpha }_{{{{\rm{sk}}}}}=({M}_{3}^{1/3}/{M}_{2}^{1/2})\), which assumes a value of 1.2 for a perfectly arranged triangular vortex lattice^{38,39}. Distortions or even melting of the vortex lattice structure, which may be caused by variations of temperature or magnetic field, are strongly reflected in α_{sk}^{40,41,42}.
Figure 2 (g) shows the temperature evolution of \({\alpha }_{{{{\rm{sk}}}}}^{  c}\) (for B_{ext}∣∣c) and \({\alpha }_{{{{\rm{sk}}}}}^{  ab}\) (for B_{ext}∣∣ab) for the kagome superconductor CsV_{3}Sb_{5}. Notably, in both directions, α_{sk}(T) remains independent of temperature for T ≲ T_{c}, with a constant value of ≃ 0.8 for \({\alpha }_{{{{\rm{sk}}}}}^{  ab}\) and 0.9 for \({\alpha }_{{{{\rm{sk}}}}}^{  c}\), respectively. We note that near the superconducting transition temperature the μSR response could be well fitted by the single Gaussian line (i.e., the reduced χ^{2} of SKG and single Gaussian fits become almost equal). Since for the symmetric P(B) distribution α_{sk} stays exactly at zero, this leads to the sudden change of α_{sk} at T ~ 2.2 K, i.e., ≃ 0.3 K below T_{c}. This observation suggests that very close to T_{c}, the FLL is slightly distorted but does not disturb the determination of the temperature evolution of the superfluid density along different crystallographic directions (as we show later), which is the main goal of the present study.
We estimate the superconducting contribution to the depolarization rate, σ_{sc}, by quadratically subtracting the temperatureindependent nuclear magnetic moment contribution σ_{nm} (obtained above T_{c}) from the the total depolarization rate σ (i.e., \({\sigma }_{{{{\rm{sc}}}}}^{2}={\sigma }^{2}{\sigma }_{{{{\rm{nm}}}}}^{2}\)).
σ_{sc} can be expressed as a function of the reduced field b = \(\frac{B}{{B}_{{{{\rm{c}}}}2}}\) (B_{c2} being the upper critical field) and the GinzburgLandau coefficient κ by the relation developed by Brandt^{43,44}
where λ is the magnetic penetration depth. Thus, we obtain the temperature dependence of \({\lambda }_{{{{\rm{ab}}}}}^{2}\) (for B_{ext}∥c) and \({\lambda }_{{{{\rm{ab,c}}}}}^{2}\) (for B_{ext}∥ab). In the case of an anisotropic superconductor, the magnetic penetration depth is also anisotropic. In the present case, by considering an anisotropic effective mass tensor, \({\lambda }_{{{{\rm{ab}}}}}^{2}\) and \({\lambda }_{{{{\rm{ab,c}}}}}^{2}\), \({\lambda }_{{{{\rm{c}}}}}^{2}\) can be estimated (see Suppl. Mater. for the detailed analysis). In this way, we can directly compare the anisotropy of the magnetic penetration depth. Figure 3a and b shows the temperature evolution of \({\lambda }_{{{{\rm{ab}}}}}^{2}\) and \({\lambda }_{{{{\rm{c}}}}}^{2}\), respectively.
As mentioned in the introduction, there are only a few studies involving different experimental techniques to address the superconducting gap structure of the CsV_{3}Sb_{5} kagome superconductor. However, there is no consensus among them. To determine whether the superconducting gap structure of this compound is of singlegap, multigap, or even of nodal nature, we analyzed the temperature dependence of magnetic penetration depth. We analyzed λ(T) data using the following expression:
where \(f={(1+E/{k}_{{{{\rm{B}}}}}T)}^{1}\) represents the Fermidistribution function. The temperature and angledependent gap function is described by Δ(T, ϕ) = Δ_{0}δ(T/T_{c})g(ϕ), where Δ_{0} is the maximum gap value at T = 0, while the temperature dependence of the gap function is^{45,46}\(\delta (T/{T}_{{{{\rm{c}}}}})=\tanh \{1.821[1.018{({T}_{{{{\rm{c}}}}}/T1)}^{0.51}]\}\). Here, g(φ) corresponds to the angular dependence of the gap and takes a value of 1 for swave and \( \cos (2\varphi )\) for dwave gap symmetry. Motivated by recent studies reporting a nodalgap structure in CsV_{3}Sb_{5}, evidenced by a nonzero value of the residual linear term of its thermal conductivity at zero field together with its rapid increase with field, we tried to fit the data using a dwave model^{29}. However, as shown in Fig. 3a and b, a dwave model cannot describe the data well. On the contrary, recent tunneling experiments^{27}, tunneldiode oscillator (TDO) based results, along with specific heat measurements conjointly indicate a multiband nature of superconductivity in CsV_{3}Sb_{5}^{28}. Thus, we proceeded to fit the \({\lambda }_{{{{\rm{c}}}}}^{2}(T)\) and \({\lambda }_{{{{\rm{ab}}}}}^{2}(T)\) data simultaneously with a twogap scenario using a weighted sum:
Here x is the weight associated with the larger gap and Δ_{0,i} (i = 1, 2 are the band indices) are the gaps related to the first and second band.
The fit of λ_{ab}(T) and λ_{c}(T) with a twogap (s + s)wave model was performed by assuming similar gap values [Δ_{1(2),ab} = Δ_{1(2),c}], but different weighting factors for the two directions (x_{ab} ≠ x_{c}). A fit with a dwave model was also performed by considering a similar gap value along two different directions. It is evident that the twogap model follows the experimental data very well [Fig. 3a and b]. Table 1 summarizes the different superconducting parameters as obtained from the fits. Note that the TFμSR measurements are well described by gap values which are very close to those obtained from tunneling measurements (Δ_{1} = 0.57 meV, Δ_{2} = 0.3 meV, Δ_{3} = 0.45 meV)^{27}. On the other hand, as shown in Fig. 3a and b, a dwave model does not describe the data well. This conclusion could be reinforced also by the systematically larger value of the reduced \({\chi }_{r}^{2}\) in case of a dwave model (1.32) compared to the s + swave model fit (1.20). In this regard, it is worthwhile mentioning that recent NMR studies on CsV_{3}Sb_{5} clearly indicate a coherence peak below T_{c}^{47}, further supporting nodeless superconductivity in CsV_{3}Sb_{5}. Although thermal conductivity measurements at ultralow temperatures^{29} suggest the presence of nodes in the superconducting energy gap, our data do not seem to be consistent with with nodal superconductivity, a result further confirmed by NMR/NQR^{47}, tunneling^{27}, specific heat, and magnetic penetration depth based on TDO measurements^{28}. Further studies are needed in order to clarify the discrepancy between the macroscopic and localprobe techniques.
Furthermore, we determined the anisotropy of the magnetic penetration depth, γ_{λ}, defined as:
By using the \({\lambda }_{{{{\rm{ab}}}}}^{2}(T)\) and \({\lambda }_{{{{\rm{c}}}}}^{2}(T)\) values, we obtain the temperature dependence of the magnetic penetration depth anisotropy as presented in Fig. 3c. According to the phenomenological GinzburgLandau theory for uniaxial anisotropic superconductors, the various anisotropies, such as the magnetic penetration depth anisotropy γ_{λ} and the upper critical field anisotropy \({\gamma }_{{B}_{{{{\rm{c}}}}2}}\), can be accounted for by a common parameter:^{48,49}
In the above equation, ξ represents the coherence length. In order to compare γ_{λ} and \({\gamma }_{{B}_{{{{\rm{c}}}}2}}\), we analyzed the electrical transport data in the presence of various applied fields and estimated the temperature dependence of the upper critical field anisotropy \({\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)\) = B_{c2,ab}(T)/B_{c2,c}(T), where B_{c2,ab}(T) and B_{c2,c}(T) are the upper critical fields corresponding to zero values of resistivity for B∥ab and B∥c (see Supplementary Fig. 1). Figure 3c depicts also the temperature dependence of \({\gamma }_{{B}_{{{{\rm{c}}}}2}}\). Interestingly, γ_{λ}(T) changes slightly from γ_{λ} ≃ 1.6 close to T_{c} to γ_{λ} ≃ 3.3 close to T = 0 K. Conversely, \({\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)\) varies strongly from \({\gamma }_{{B}_{{{{\rm{c}}}}2}}\simeq 7.1\) close to T_{c} to \({\gamma }_{{B}_{{{{\rm{c}}}}2}}\simeq 8.5\) when T_{c} ≃ 0.3 K.
ZFμSR experiment in the superconducting state
Finally, to check whether superconductivity breaks timereversal symmetry in CsV_{3}Sb_{5}, we performed zerofield (ZF) μSR measurements at different temperatures across T_{c}. A timereversal symmetrybreaking superconducting state is one of the hallmarks of certain unconventional superconductors; in particular, in many Rebased superconductors^{50}, Sr_{2}RuO_{4}^{51,52,53}, Ba_{1−x}K_{x}Fe_{2}As_{2}^{54}, LaNiC_{2}^{55}, La_{7}Ir_{3}^{56} etc. Due to the high muon sensitivity to low magnetic fields, μSR has been the technique of choice for studying it. Here, the ZFμSR spectra are well described by a damped Gaussian KuboToyabe depolarization function^{57} (see Supp. Mater. for a detailed description of analysis), which considers the field distribution at the muon site created by both nuclear and electronic moments. The (ZF) μSR experiments were carried out with the muon spin rotated by ~43^{∘} with respect to the direction of muon momentum. This configuration allows us to use two different detector pairs, namely, Forward(F) Backward(B) and Left(L)Right(R), to probe the anisotropic ZF response (see Supplementary Fig. 2 for a schematic view of the experimental geometry). Figure 4a and b shows the ZFμSR spectra collected above and below T_{c}. Figure 4c shows the temperature dependence of the electronic relaxation rates for different detectors. The shaded regions represent the statistical scattering of data points. It is evident from Fig. 4c that the relaxation rates do not show any noticeable increase below T_{c}, indicating that superconductivity does not break TRS in CsV_{3}Sb_{5} within the statistical accuracy, similar to the recent results on the related system KV_{3}Sb_{5}^{25}. Further ZF experiments with high statistics would be interesting in order to confirm or disprove the presence of time reversal symmetry breaking in the superconducting ground state.
DISCUSSION
The probability field distribution determined experimentally shows a highly asymmetric lineshape, indicative of a wellordered FLL in the vortex state of the superconductor CsV_{3}Sb_{5}, and it remains almost independent of temperature until very close to T_{c}. This observation unambiguously suggests that the FLL in CsV_{3}Sb_{5} is well arranged in the superconducting state and it gets slightly distorted only in the vicinity of T_{c}. In general, the change of α_{sk} as a function of magnetic field and temperature is associated with the vortex lattice melting^{40,41,58}, and/or a dimensional crossover from a threedimensional (3D) to a twodimensional (2D) type of FLL^{39,41}. Both processes are thermally activated and caused by increased vortex mobility via a loosening of the inter or intraplanar FLL correlations^{39}. Another possibility involves the rearrangement of the vortex lattice induced by a change of the anisotropy coefficient γ_{λ} = λ_{c}/λ_{ab}^{59}. Since CsV_{3}Sb_{5} has a very small superconducting anisotropy (γ_{λ} ≃ 3, see below), we can rule out a possible vortexmelting scenario. As for the anisotropyinduced FLL rearrangement, the temperature evolution of α_{sk} measured in B_{ext}∥ab and B_{ext}∥c experiments are expected to be very much different^{59}, which is also not the case here [see Fig. 3c]. Therefore, we are left with the explanation that close to T_{c}, where the broadening of μSR signal caused by formation of FLL becomes comparable with or even smaller than the relaxation caused by the nuclear magnetic moments [straight lines in Fig. 2e], the shape of P(B) distribution is dominated by the symmetric ‘nuclear’ term, which effectively pushes α_{sk} to zero shortly before the superconducting transition temperature T_{c} is reached.
Furthermore, a detailed analysis of the λ(T) data reveals the presence of two superconducting gaps at the Fermi surface, with gap values of 0.6 and 0.23 meV. This conclusion is in agreement with recent reports involving different experimental techniques^{28,60}. As μSR is a bulk probe, we conclude that the bulk superconducting gap of this compound consists of two swave gaps rather than a nodal gap. Another interesting observation is the fact that the T_{c}/λ^{−2}(0) ratio for CsV_{3}Sb_{5} in both field orientations is comparable to those of hightemperature unconventional superconductors and ironpnictides^{61,62}. Systems with a small T_{c}/λ^{−2}(0) ~ 0.00025–0.015 are usually considered to be BCSlike, while large T_{c}/λ^{−2}(0) values are expected only in the BEClike picture and is considered a hallmark feature of unconventional superconductivity. This approach has become a key feature to characterize BCSlike (socalled conventional) and BEClike superconductors. Remarkably, in CsV_{3}Sb_{5}, T_{c}/λ^{−2}(0) is as high as ~ 0.2 (for \({\lambda }_{{{{\rm{ab}}}}}^{2}(0)\))  2.2 (\({\lambda }_{{{{\rm{c}}}}}^{2}(0)\)), where the lower limit is comparable to the unconventional transition metal dichalcogenide superconductors^{63} and the upper limit is close to T_{c}/λ^{−2}(0) ~ 4 of holedoped cuprates^{61,62,64}. This point towards an unconventional pairing mechanism in the kagome superconductor CsV_{3}Sb_{5}.
Moreover, we observe a clear difference in the temperature dependence of the anisotropies related to magnetic penetration depth and upper critical fields, again signaling a clear deviation from the GinzburgLandau theory. This situation finds a clear parallel with the data from the wellknown twogap superconductor MgB_{2}^{65,66}, Febased Sm and Nd1111 systems^{67,68}, 122 pnictide superconductors: Ba(Fe_{1−x}Co_{x})_{2}As_{2}^{69,70}, (Ba_{1−x}K_{x})Fe_{2}As_{2}^{59}; FeSe_{0.5}Te_{0.5}^{71}, CaKFe_{4}As_{4}^{72}, etc., where two different temperature variations of γ_{λ}(T) and \({\gamma }_{{B}_{{{{\rm{c}}}}2}}(T)\) were attributed to multiband superconductivity. Thus, in comparison to wellestablished multigap superconductors, we find further support for the multigap behavior in this compound. Therefore, our results provide microscopic evidence of anisotropic multigap superconductivity in the kagome superconductor CsV_{3}Sb_{5} and encourage further theoretical and experimental research on the kagome superconductors.
Finally, ZFμSR experiments suggest that across T_{c}, the relaxation rates do not change. This suggests that within the statistical accuracy, no clear signature of TRS breaking is observed in the superconducting state of CsV_{3}Sb_{5}.
METHODS
Sample preparation
Single crystals of CsV_{3}Sb_{5} were grown from Cs ingots (purity 99.9%), V 3N powder (purity 99.9%) and Sb grains (purity 99.999%) using the selfflux method, similar to the growth of RbV_{3}Sb_{5}^{73}. The eutectic mixture of CsSb and CsSb_{2} was mixed with VSb_{2} to form a composition with approximately 50 at% Cs_{x}Sb_{y} and 50 at% VSb_{2}. The mixture was put into an alumina crucible and sealed in a quartz ampoule under partial argon atmosphere. The sealed quartz ampoule was heated to 1273 K in 12 h and kept there for 24 h. Then it was cooled down to 1173 K at 50 K/h and further to 923 K at a slower rate. Finally, the ampoule was removed from the furnace and decanted with a centrifuge to separate the CsV_{3}Sb_{5} crystals from the flux. The obtained crystals have a typical size of 4 × 4 × 1 mm^{3} and are stable in air over a period of at least several months. As shown in Fig. 1a and b, the fluxgrown single crystals possess an obvious hexagonal symmetry, while the Xray Laue diffraction images demonstrate the single crystallinity of the material. The XRD pattern was collected using a Bruker D8 xray diffractometer with Cu K_{α} radiation (λ = 0.15418 nm) at room temperature.
Magnetic susceptibility and electrical transport measurements
The magnetization measurements were performed in a Quantum Design magnetic property measurement system SQUID magnetometer under fieldcooled and zerofieldcooled conditions. Electrical transport measurements were carried out in a Quantum Design physical property measurement system (PPMS14T). The longitudinal electrical resistivity was measured using a fourprobe method with the current flowing in the ab plane. For measurements in the temperature range 0.3–2 K we employed a Heliox recondensing He3 system, which is equipped with a superconducting NbTi twisted pair cable for the measurements.
μSRexperiments
We performed transverse field (TF) and zero field μSR experiments using the Dolly spectrometer (πE1 beamline) at the Paul Scherrer Institute (Villigen, Switzerland). Since the crystals were rather thick (~1 mm), they were mounted in a single layer using Apiezon N grease, to form a mosaic covering an area of 7 × 7 mm^{2}. The Dolly spectrometer is equipped with a standard veto setup, providing a lowbackground μSR signal. All TF μSR experiments were done after fieldcooling the sample with the applied field either along the kagome plane or perpendicular to it. The μSR time spectra were analyzed using the open software package MUSRFIT^{74}.
Notes added
While preparing this manuscript, we came to know about another μSR work studying the high temperature charge ordered phase of CsV_{3}Sb_{5} [arXiv:2107.10714 (2021) (https://arxiv.org/abs/2107.10714)].
Data availability
The data supporting the findings of this study are available within the paper and in the Supplementary Information. The raw data are available from the corresponding authors upon reasonable request or raw data can also be found at the following link http://musruser.psi.ch/cgibin/SearchDB.cgi.
References
Yin, J.X. et al. Negative flatband magnetism in a spinorbit coupled kagome magnet. Nat. Phys. 15, 443 (2019).
Guguchia, Z. et al. Tunable anomalous Hall conductivity through volumewise magnetic competition in a topological kagome magnet. Nat. Commun. 11, 559 (2020).
Ghimire, N. & Mazin, I. Topology and correlations on the kagome lattice. Nat. Mater. 19, 137–138 (2020).
Ye, L. et al. Massive Dirac fermions in a ferromagnetic kagome metal. Nature 555, 638–642 (2018).
Cao, Y. et al. Correlated insulator behaviour at halffilling in magicangle graphene superlattices. Nature 556, 80–84 (2018).
Hofmann, J. S., Berg, E. & Chowdhury, D. Superconductivity, pseudogap, and phase separation in topological flat bands. Phys. Rev. B 102, 201112(R) (2020).
Nunes, L. H. C. M. & Smith, C. M. Flatband superconductivity for tightbinding electrons on a squareoctagon lattice. Phys. Rev. B 101, 224514 (2020).
Mielke III, C. et al. Nodeless kagome superconductivity in LaRu_{3}Si_{2}. Phys. Rev. Mat. 5, 034803 (2021).
Ortiz, B. et al. New kagome prototype materials: discovery of KV_{3}Sb_{5}, RbV_{3}Sb_{5}, and CsV_{3}Sb_{5}. Phys. Rev. Mat. 3, 09447 (2019).
Yang, S. et al. Giant, unconventional anomalous Hall effect in the metallic frustrated magnet candidate, KV_{3}Sb_{5}. Sci. Adv. 6, 1–7s (2020).
Yu, F. et al. Concurrence of anomalous Hall effect and charge density wave in a superconducting topological kagome metal. Phys. Rev. B 104, L041103 (2021).
Zhao, H. et al. Cascade of correlated electron states in a kagome superconductor CsV_{3}Sb_{5}. Nature 599, 216–221 (2021).
Fu, Y. et al. Quantum transport evidence of topological band structures of kagome superconductor CsV_{3}Sb_{5}. Phys. Rev. Lett. 127, 207002 (2021).
Jiang, Y.X. et al. Unconventional chiral charge order in kagome superconductor KV_{3}Sb_{5}. Nat. Mater. 20, 1353–1357 (2021).
Denner, M., Thomale, R. & Neupert., T. Analysis of charge order in the kagome metal AV_{3}Sb_{5} (A = K, Rb, Cs). Phys. Rev. Lett. 127, 217601 (2021).
Shumiya, N. et al. Intrinsic nature of chiral charge order in the kagome superconductor RbV_{3}Sb_{5}. Phys. Rev. B 104, 035131 (2021).
Wang, Z. et al. Electronic nature of chiral charge order in the kagome superconductor CsV_{3}Sb_{5}. Phys. Rev. B 104, 075148 (2021).
Kang, M. et al. Twofold van Hove singularity and origin of charge order in topological kagome superconductor CsV_{3}Sb_{5}. Nat. Phys. https://doi.org/10.1038/s41567021014515 (2022).
Li, H. et al. Observation of unconventional charge density wave without acoustic phonon anomaly in kagome superconductors AV_{3}Sb_{5} (A = Rb, Cs). Phys. Rev. X 11, 031050 (2021).
Song, D. et al. Orbital ordering and fluctuations in a kagome superconductor CsV_{3}Sb_{5}. Sci. China: Phys. Mech. Astron. 65, 247462 (2022).
Ortiz, B. et al. CsV_{3}Sb_{5}: A Z_{2} topological kagome metal with a superconducting ground state. Phys. Rev. Lett. 125, 247002 (2020).
Ortiz, B. et al. Superconductivity in the Z_{2} kagome metal KV_{3}Sb_{5}. Phys. Rev. Mater. 5, 034801 (2021).
Yin, Q. et al. Superconductivity and normalstate properties of kagome metal RbV_{3}Sb_{5} single crystals. Chin. Phys. Lett. 38, 037403 (2021).
Liu, Z. et al. Chargedensitywaveinduced bands renormalization and energy gaps in a kagome superconductor RbV_{3}Sb_{5}. Phys. Rev. X 11, 041010 (2021).
Mielke III, C. et al. Timereversal symmetrybreaking charge order in a correlated kagome superconductor. Nature 602, 245 (2022).
Wang, Y. et al. Proximityinduced spintriplet superconductivity and edge supercurrent in the topological kagome metal, KV_{1−x}Sb_{5}. Preprint at https://arxiv.org/abs/2012.05898 (2020).
Xu, H.S. et al. Multiband superconductivity with signpreserving order parameter in kagome superconductor CsV_{3}Sb_{5}. Phys. Rev. Lett. 127, 187004 (2021).
Duan, W. et al. Nodeless superconductivity in the kagome metal CsV_{3}Sb_{5}. Sci. ChinaPhys. Mech. Astron. 64, 107462 (2021).
Zhao, C. et al. Nodal superconductivity and superconducting domes in the topological kagome metal CsV_{3}Sb_{5}. Preprint at https://arxiv.org/abs/2102.08356 (2021).
Zhang, Z. et al. Pressureinduced reemergence of superconductivity in the topological kagome metal CsV_{3}Sb_{5}. Phys. Rev. B 103, 224513 (2021).
Chen, X. et al. Highlyrobust reentrant superconductivity in CsV_{3}Sb_{5} under pressure. Chin. Phys. Lett. 38, 057402 (2021).
Chen, K. et al. Double superconducting dome and triple enhancement of T_{c} in the kagome superconductor CsV_{3}Sb_{5} under high pressure. Phys. Rev. Lett. 126, 247001 (2021).
Wu, X. et al. Nature of unconventional pairing in the kagome superconductors AV_{3}Sb_{5} (A = K, Rb, Cs). Phys. Rev. Lett. 127, 177001 (2021).
Kiesel, M. L. & Thomale, R. Sublattice interference in the kagome Hubbard model. Phys. Rev. B 86, 121105 (2012).
Sonier, J. E., Brewer, J. H. & Kiefl, R. F. μSR studies of the vortex state in typeII superconductors. Rev. Mod. Phys. 72, 769 (2000).
Ouladdiaf, B. et al. OrientExpress: a new system for Laue neutron diffraction. Phys. B: Condens. Matter. 385386, 1052 (2006).
Maisuradze, A., Khasanov, R., Shengelaya, A. & Keller, A. Comparison of different methods for analyzing μSR line shapes in the vortex state of typeII superconductors. J. Phys.: Condens. Matter 21, 075701 (2009).
Thiemann, S., Radović, Z. & Kogan, V. Field structure of vortex lattices in uniaxial superconductors. Phys. Rev. B 39, 11406 (1989).
Aegerter, C. et al. Angular dependence of the disorder crossover in the vortex lattice of Bi_{2.15}Sr_{1.85}CaCu_{2}O_{8+δ} by muon spin rotation and torque magnetometry. Phys. Rev. B 57, 1253 (1998).
Lee, S. et al. Evidence for fluxlattice melting and a dimensional crossover in singlecrystal Bi_{2.15}Sr_{1.85}CaCu_{2}O_{8+δ} from muon spin rotation studies. Phys. Rev. Lett. 71, 3862 (1993).
Blasius, T. et al. Evidence for a twostage melting transition of the vortex matter in Bi_{2}Sr_{2}Ca_{1}Cu_{2}O_{8+δ} single crystals obtained by muon spin rotation. Phys. Rev. Lett. 82, 4926 (1999).
Heron, D. O. G. et al. Muonspin rotation measurements of an unusual vortexglass phase in the layered superconductor Bi_{2.15}Sr_{1.85}CaCu_{2}O_{8+δ}. Phys. Rev. Lett. 110, 107004 (2013).
Brandt, E. H. Flux distribution and penetration depth measured by muon spin rotation in highT_{c} superconductors. Phys. Rev. B 37, 2349(R) (1988).
Brandt, E. H. Properties of the ideal GinzburgLandau vortex lattice. Phys. Rev. B 68, 054506 (2003).
Khasanov, R., Gupta, R., Das, D., LeitheJasper, A. & Svanidze, E. Singlegap versus twogap scenario: Specific heat and thermodynamic critical field of the noncentrosymmetric superconductor BeAu. Phys. Rev. B 102, 014514 (2020).
Carrington, A. & Manzano, F. Magnetic penetration depth of MgB_{2}. Physica C 385, 205 (2003).
Mu, C. et al. Swave superconductivity in kagome metal CsV_{3}Sb_{5} revealed by ^{121/123}Sb NQR and ^{51}V NMR measurements. Chin. Phys. Lett. 38, 077402 (2021).
Kogan, V. G. London approach to anisotropic typeII superconductors. Phys. Rev. B 24, 1572 (1981).
Tinkham, M. Introduction to Superconductivity. Krieger Publishing Company, Malabar, Florida, (1975).
Shang, T. & Shiroka, T. Timereversal symmetry breaking in Rebased superconductors: recent developments. Front. Phys. 9, 270 (2021).
Luke, G. Timereversal symmetrybreaking superconductivity in Sr_{2}RuO_{4}. Nature 394, 559 (1998).
Grinenko, V. Unsplit superconducting and time reversal symmetry breaking transitions in Sr_{2}RuO_{4} under hydrostatic pressure and disorder. Nat. Commun. 12, 3920 (2021).
Grinenko, V. Split superconducting and timereversal symmetrybreaking transitions in Sr_{2}RuO_{4} under stress. Nat. Phys. 17, 748 (2021).
Grinenko, V. Superconductivity with broken timereversal symmetry inside a superconducting swave state. Nat. Phys. 16, 789 (2020).
Hillier, A. D., Quintanilla, J. & Cywinski, R. Evidence for timereversal symmetry breaking in the noncentrosymmetric superconductor LaNiC_{2}. Phys. Rev. Lett. 102, 117007 (2009).
Barker, J. A. T. et al. Unconventional superconductivity in La_{7}Ir_{3} revealed by muon spin relaxation: introducing a new family of noncentrosymmetric superconductor that breaks timereversal symmetry. Phys. Rev. Lett. 115, 267001 (2015).
Hayano, R. et al. Zero and lowfield spin relaxation studied by positive muons. Phys. Rev. B 20, 850 (1979).
Khasanov, R. et al. Evidence for a competition between the superconducting state and the pseudogap state of (BiPb)_{2}(SrLa)_{2}CuO_{6+δ} from muon spin rotation experiments. Phys. Rev. Lett. 101, 227002 (2008).
Khasanov, R. et al. Twogap superconductivity in Ba_{1−x}K_{x}Fe_{2}As_{2}: A complementary study of the magnetic penetration depth by muonspin rotation and angleresolved photoemission. Phys. Rev. Lett. 102, 187005 (2009).
Ni, S. et al. Anisotropic superconducting properties of kagome metal CsV_{3}Sb_{5}. Chin. Phys. Lett. 38, 057403 (2021).
Uemura, Y. et al. Universal correlations between T_{c} and n_{s}/m (carrier density over effective mass) in highT_{c} cuprate superconductors. Phys. Rev. Lett. 62, 2317 (1989).
Uemura, Y. et al. Basic similarities among cuprate, bismuthate, organic, Chevrelphase, and heavyfermion superconductors shown by penetrationdepth measurements. Phys. Rev. Lett. 66, 2665 (1991).
Guguchia, Z. et al. Signatures of the topological s^{+−} superconducting order parameter in the typeII weyl semimetal T_{d}MoTe_{2}. Nat. Comm. 8, 1082 (2017).
Gupta, R., Ying, T. P., Qi, Y. P., Hosono, H. & Khasanov, R. Gap symmetry of the noncentrosymmetric superconductor W_{3}Al_{2}C. Phys. Rev. B 103, 174511 (2021).
Angst, M. et al. Temperature and field dependence of the anisotropy of MgB_{2}. Phys. Rev. Lett. 88, 167004 (2002).
Fletcher, J. D., Carrington, A., Taylor, O. J., Kazakov, S. M. & Karpinski, J. Temperaturedependent anisotropy of the penetration depth and coherence length of MgB_{2}. Phys. Rev. Lett. 95, 097005 (2005).
Weyeneth, S. et al. Evidence for two dstinct anisotropies in the oxypnictide superconductors SmFeAsO_{0.8}F_{0.2} and NdFeAsO_{0.8}F_{0.2}. Supercond. Nov. Magn. 22, 347–351 (2009).
Gupta, R. et al. Selfconsistent twogap approach in studying multiband superconductivity of NdFeAsO_{0.65}F_{0.35}. Front. Phys. 8, 2 (2020).
Prozorov, R. Anisotropic London penetration depth and superfluid density in single crystals of ironbased pnictide superconductors. Physica C 469, 582–589 (2009).
Tanatar, M. A. et al. Anisotropy of the iron pnictide superconductor Ba(Fe_{1−x}Co_{x})_{2}As_{2} (x = 0.074, T_{c} = 23 K). Phys. Rev. B 79, 094507 (2009).
Bendele, M. et al. Anisotropic superconducting properties of singlecrystalline FeSe_{0.5}Te_{0.5}. Phys. Rev. B 81, 224520 (2010).
Khasanov, R. et al. Anisotropy induced vortex lattice rearrangement in CaKFe_{4}As_{4}. Phys. Rev. B 99, 140507(R) (2019).
Yin, Q. W. et al. Superconductivity and normalstate properties of kagome metal RbV_{3}Sb_{5} single crystals. Chin. Phys. Lett. 38, 037403 (2021).
Suter, A. & Wojek, B. M. Musrfit: a free platformindependent framework for μSR data analysis. Phys. Procedia 30, 69 (2012).
Yaouanc, A., Dalmas de Reotier, P. & Brandt, E. H. Effect of the vortex core on the magnetic field in hard superconductors. Phys. Rev. B 55, 11107 (1997).
Acknowledgements
μSR experiments were performed at the Swiss Muon Source (SμS), Paul Scherrer Institute (PSI), Switzerland. The work of RG was supported by the Swiss National Science Foundation (SNFGrant No. 200021175935). H.C.L. was supported by National Natural Science Foundation of China (Grant Nos. 11822412 and 11774423), Ministry of Science and Technology of China (Grant Nos. 2018YFE0202600 and 2016YFA0300504) and the Beijing Natural Science Foundation (Grant No. Z200005).
Author information
Authors and Affiliations
Contributions
R.G., D.D., and C.M. contributed equally to this work. R.G., D.D., C.M., Z.G., R.K., and C.B. carried out μSR experiments. R.G., D.D., C.M., and R.K. performed μSR data analysis. Q.Y., Z.T., C.G., and H.C.L. provided and characterized samples. M.B. performed low temperature electrical transport measurements under different fields. H.L. supervised the work at PSI. R.G., D.D., C.M., Z.G., and T.S. prepared the manuscript with notable inputs from all authors.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gupta, R., Das, D., Mielke III, C.H. et al. Microscopic evidence for anisotropic multigap superconductivity in the CsV_{3}Sb_{5} kagome superconductor. npj Quantum Mater. 7, 49 (2022). https://doi.org/10.1038/s41535022004537
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41535022004537
This article is cited by

Bulk evidence of anisotropic swave pairing with no sign change in the kagome superconductor CsV3Sb5
Nature Communications (2023)

Chiral excitonic order from twofold van Hove singularities in kagome metals
Nature Communications (2023)

Tunable unconventional kagome superconductivity in charge ordered RbV3Sb5 and KV3Sb5
Nature Communications (2023)

Superconducting gap evolution of kagome metal CsV3Sb5 under pressure
Science China Physics, Mechanics & Astronomy (2023)

Topological kagome magnets and superconductors
Nature (2022)